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xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="fields_and_quanta">Fields and quanta</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> and <a class="existingWikiWord" href="/nlab/show/fundamental+particle">particles</a> in <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> and in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>: <a class="existingWikiWord" href="/nlab/show/force">force</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/photon">photon</a> - <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> (<a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/W-boson">W</a>, <a class="existingWikiWord" href="/nlab/show/Z-boson">Z</a>, <a class="existingWikiWord" href="/nlab/show/B-boson">B-boson</a> - <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak field</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/gluon">gluon</a> - <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infraparticle">infraparticle</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Higgs+boson">Higgs boson</a>, <a class="existingWikiWord" href="/nlab/show/inflaton">inflaton</a> (<a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>)</li> </ul> <a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> (<a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>, <a class="existingWikiWord" href="/nlab/show/Dirac+fields">Dirac fields</a>) <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a></td><td style="text-align: left;">1st generation</td><td style="text-align: left;">2nd generation</td><td style="text-align: left;">3d generation</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">up-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/up+quark">up quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/charm+quark">charm quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/top+quark">top quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;">down-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/down+quark">down quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">charged</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron">electron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon">muon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tauon">tauon</a></td></tr> <tr><td style="text-align: left;">neutral</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron+neutrino">electron neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon+neutrino">muon neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tau+neutrino">tau neutrino</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a>: <a class="existingWikiWord" href="/nlab/show/pion">pion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons: <a class="existingWikiWord" href="/nlab/show/%CF%95-meson">ϕ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mover><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">s \bar s</annotation></semantics></math>), <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">d s</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>u</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u u + d d + s s</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a>: <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> u c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">s c</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/J%2F%CF%88-meson">J/ψ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">c \bar c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a>: <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">q b</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mover><mi>b</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">b \bar b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a>: <a class="existingWikiWord" href="/nlab/show/proton">proton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>u</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u u d)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u d d)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> (also: <a class="existingWikiWord" href="/nlab/show/antiparticles">antiparticles</a>) <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective particles</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/hadrons">hadrons</a> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of the above <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>) <ul> <li> <a class="existingWikiWord" href="/nlab/show/meson">meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/scalar+meson">scalar meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%CF%83-meson">σ-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pion">pion</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/b1-meson">b1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/h1-meson">h1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+meson">tensor meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quarkonium">quarkonium</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/charmonium">charmonium</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/exotic+meson">exotic meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/XYZ+meson">XYZ meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tetraquark">tetraquark</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/baryon">baryon</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/proton">proton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/chemical+element">chemical element</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/carbon">carbon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/nitrogen">nitrogen</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lambda+baryon">Lambda baryon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pentaquark">pentaquark</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/solitons">solitons</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/caloron">caloron</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> </ul> in <a class="existingWikiWord" href="/nlab/show/grand+unified+theory">grand unified theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/leptoquark">leptoquark</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Z%27-boson">Z'-boson</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/MSSM">minimally extended supersymmetric standard model</a> <a class="existingWikiWord" href="/nlab/show/superpartners">superpartners</a> bosinos: <ul> <li> <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> (<a class="existingWikiWord" href="/nlab/show/Rarita-Schwinger+field">Rarita-Schwinger field</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/gaugino">gaugino</a> - <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gluino">gluino</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/sfermions">sfermions</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/squark">squark</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a> candidates <ul> <li> <a class="existingWikiWord" href="/nlab/show/WIMP">WIMP</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/axion">axion</a> </li> </ul> Exotica <ul> <li> <a class="existingWikiWord" href="/nlab/show/preon">preon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/graviphoton">graviphoton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monopole">monopole</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dual+graviton">dual graviton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/giant+graviton">giant graviton</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/auxiliary+fields">auxiliary fields</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/ghost+field">ghost field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/antifield">antifield</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/antighost+field">antighost field</a>, <a class="existingWikiWord" href="/nlab/show/Nakanishi-Lautrup+field">Nakanishi-Lautrup field</a> </li> </ul> </div></div> <h4 id="qunantum_field_theory">Qunantum Field Theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>) <a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a> <h2 id="concepts">Concepts</h2> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+history">field history</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> </li> </ul> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>, <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+net">field net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> <a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>) </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#original_articles'>Original articles</a></li> <li><a href='#review'>Review</a></li> <li><a href='#phenomenology'>Phenomenology</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> Quantum electrodynamics (“QED”) is the <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> coupled to a <a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a> via the <a class="existingWikiWord" href="/nlab/show/electron-photon+interaction">electron-photon interaction</a>: it describes the <a class="existingWikiWord" href="/nlab/show/quantum+theory">quantum theory</a> of <a class="existingWikiWord" href="/nlab/show/photons">photons</a> and <a class="existingWikiWord" href="/nlab/show/electrons">electrons</a>. <h2 id="definition">Definition</h2> The <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> defining QED is the <a class="existingWikiWord" href="/nlab/show/sum">sum</a> of <ol> <li> the <a class="existingWikiWord" href="/nlab/show/free+field">free field</a> contribution of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/free+field">free field</a> contribution of a <a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a>; </li> <li> the <a class="existingWikiWord" href="/nlab/show/electron-photon+interaction">electron-photon interaction</a> term with <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a> given by the <a class="existingWikiWord" href="/nlab/show/fine+structure+constant">fine structure constant</a>. </li> </ol> See at <a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">A first idea of quantum field theory</a> <a href="A+first+idea+of+quantum+field+theory#LagrangianQED">this example</a>. The underlying <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a> admits a <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> via the corresponding <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a>. QED is, by definition, the <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> obtained from this by finding a perturbative <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> for the <a class="existingWikiWord" href="/nlab/show/electron-photon+interaction">electron-photon interaction</a>. As a result, <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> for <a class="existingWikiWord" href="/nlab/show/electron-photon+interactions">electron-photon interactions</a> in QED are then expressed in terms of <a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a>: <div> <div class="num_example"> <h6 id="example">Example</h6> (<a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> – example of <a class="existingWikiWord" href="/nlab/show/QED">QED</a>) In <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> are labeled <a class="existingWikiWord" href="/nlab/show/multigraphs">multigraphs</a> that encode <a class="existingWikiWord" href="/nlab/show/product+of+distributions">products of</a> <a class="existingWikiWord" href="/nlab/show/Feynman+propagators">Feynman propagators</a>, called <a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> (<a href="S-matrix#FeynmanPerturbationSeriesAwayFromCoincidingPoints">this prop.</a>) which in turn contribute to <a class="existingWikiWord" href="/nlab/show/probability+amplitudes">probability amplitudes</a> for physical <a class="existingWikiWord" href="/nlab/show/scattering">scattering</a> processes – <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a>: The <a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> are the summands in the <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a>-expansion of the <a class="existingWikiWord" href="/nlab/show/scattering+matrix">scattering matrix</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_1"><semantics><mrow><mi>𝒮</mi><mrow><mo>(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>)</mo></mrow><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>i</mi><mi>ℏ</mi><msup><mo stretchy="false">)</mo> <mi>k</mi></msup></mrow></mfrac><mi>T</mi><mo stretchy="false">(</mo><munder><munder><mrow><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>S</mi> <mi>int</mi></msub></mrow><mo>⏟</mo></munder><mrow><mi>k</mi><mspace width="thinmathspace"></mspace><mtext>factors</mtext></mrow></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{S} \left( S_{int} \right) = \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} \frac{1}{(i \hbar)^k} T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots , S_{int}}} ) </annotation></semantics></math></div> of a given <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_2"><semantics><mrow><msub><mi>L</mi> <mi>int</mi></msub></mrow><annotation encoding="application/x-tex">L_{int}</annotation></semantics></math>. The Feynman amplitudes are the summands in an expansion of the <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_3"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(\cdots)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> with itself, which, away from coincident vertices, is given by the <a class="existingWikiWord" href="/nlab/show/star+product">star product</a> of the <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_4"><semantics><mrow><msub><mi>Δ</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_F</annotation></semantics></math> (<a href="S-matrix#TimeOrderedProductAwayFromDiagonal">this prop.</a>), via the <a class="existingWikiWord" href="/nlab/show/exponential">exponential</a> contraction <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_5"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>,</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>prod</mi><mo>∘</mo><mi>exp</mi><mrow><mo>(</mo><mi>ℏ</mi><mo>∫</mo><msubsup><mi>Δ</mi> <mi>F</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><msup><mstyle mathvariant="bold"><mi>Φ</mi></mstyle> <mi>a</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⊗</mo><mfrac><mi>δ</mi><mrow><mi>δ</mi><mstyle mathvariant="bold"><mi>Φ</mi></mstyle><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>int</mi></msub><mo>⊗</mo><msub><mi>S</mi> <mi>int</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T(S_{int}, S_{int}) \;=\; prod \circ \exp \left( \hbar \int \Delta_{F}^{a b}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}(y)} \right) ( S_{int} \otimes S_{int} ) \,. </annotation></semantics></math></div> Each <a class="existingWikiWord" href="/nlab/show/edge">edge</a> in a <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a> corresponds to a factor of a <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_6"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><munder><munder><mrow><msub><mi>S</mi> <mi>int</mi></msub><mi>⋯</mi><msub><mi>S</mi> <mi>int</mi></msub></mrow><mo>⏟</mo></munder><mrow><mi>k</mi><mspace width="thinmathspace"></mspace><mtext>factors</mtext></mrow></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T( \underset{k \, \text{factors}}{\underbrace{S_{int} \cdots S_{int}}} )</annotation></semantics></math>, being a <a class="existingWikiWord" href="/nlab/show/distribution+of+two+variables">distribution of two variables</a>; and each <a class="existingWikiWord" href="/nlab/show/vertex">vertex</a> corresponds to a factor of the <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_7"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math>. For example <a class="existingWikiWord" href="/nlab/show/quantum+electrodynamics">quantum electrodynamics</a> in <a class="existingWikiWord" href="/nlab/show/Gaussian-averaged+Lorenz+gauge">Gaussian-averaged Lorenz gauge</a> involves (via <a href="S-matrix#FieldSpeciesQED">this example</a>): <ol> <li> the <a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a> modelling the <a class="existingWikiWord" href="/nlab/show/electron">electron</a>, with <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> called the <a class="existingWikiWord" href="/nlab/show/electron+propagator">electron propagator</a> (<a href="Feynman+propagator#FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime">this def.</a>), here to be denoted <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_8"><semantics><mrow><mi>Δ</mi><mphantom><mi>AAAA</mi></mphantom><mtext>electron propagator</mtext></mrow><annotation encoding="application/x-tex"> \Delta \phantom{AAAA} \text{electron propagator} </annotation></semantics></math></div></li> <li> the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> modelling the <a class="existingWikiWord" href="/nlab/show/photon">photon</a>, with <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> called the <a class="existingWikiWord" href="/nlab/show/photon+propagator">photon propagator</a> (<a href="A+first+idea+of+quantum+field+theory#PhotonPropagatorInGaussianAveragedLorenzGauge">this prop.</a>), here to be denoted <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_9"><semantics><mrow><mi>G</mi><mphantom><mi>AAAA</mi></mphantom><mtext>photon propagator</mtext></mrow><annotation encoding="application/x-tex"> G \phantom{AAAA} \text{photon propagator} </annotation></semantics></math></div></li> <li> the <a class="existingWikiWord" href="/nlab/show/electron-photon+interaction">electron-photon interaction</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_10"><semantics><mrow><msub><mi>L</mi> <mi>int</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><mi>i</mi><mi>g</mi><mo stretchy="false">(</mo><msup><mi>γ</mi> <mi>μ</mi></msup><msup><mo stretchy="false">)</mo> <mi>α</mi></msup><msub><mrow></mrow> <mi>β</mi></msub></mrow><mo>⏟</mo></munder><mtext>interaction</mtext></munder><mspace width="thinmathspace"></mspace><munder><munder><mover><mrow><msub><mi>ψ</mi> <mi>α</mi></msub></mrow><mo>¯</mo></mover><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mtext>incoming</mtext></mrow><mrow><mtext>electron</mtext></mrow></mfrac></mrow><mrow><mtext>field</mtext></mrow></mfrac></munder><mspace width="thickmathspace"></mspace><munder><munder><mrow><msub><mi>a</mi> <mi>μ</mi></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mtext>photon</mtext></mrow></mfrac></mrow><mrow><mtext>field</mtext></mrow></mfrac></munder><mspace width="thickmathspace"></mspace><munder><munder><mrow><msup><mi>ψ</mi> <mi>β</mi></msup></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mtext>outgoing</mtext></mrow><mrow><mtext>electron</mtext></mrow></mfrac></mrow><mrow><mtext>field</mtext></mrow></mfrac></munder></mrow><annotation encoding="application/x-tex"> L_{int} \;=\; \underset{ \text{interaction} }{ \underbrace{ i g (\gamma^\mu)^\alpha{}_\beta } } \, \underset{ { \text{incoming} \atop \text{electron} } \atop \text{field} }{\underbrace{\overline{\psi_\alpha}}} \; \underset{ { \, \atop \text{photon} } \atop \text{field} }{\underbrace{a_\mu}} \; \underset{ {\text{outgoing} \atop \text{electron} } \atop \text{field} }{\underbrace{\psi^\beta}} </annotation></semantics></math></div></li> </ol> The <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a> for the <a class="existingWikiWord" href="/nlab/show/electron-photon+interaction">electron-photon interaction</a> alone is <center> <img src="https://ncatlab.org/nlab/files/InteractionVertexOfQED.jpg" width="150" /> </center> where the solid lines correspond to the <a class="existingWikiWord" href="/nlab/show/electron">electron</a>, and the wiggly line to the <a class="existingWikiWord" href="/nlab/show/photon">photon</a>. The corresponding <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is (written in <a class="existingWikiWord" href="/nlab/show/generalized+function">generalized function</a>-notation) <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="block" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_11"><semantics><mrow><munder><munder><mrow><msup><mi>ℏ</mi> <mrow><mn>3</mn><mo stretchy="false">/</mo><mn>2</mn><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>⏟</mo></munder><mtext>loop order</mtext></munder><munder><munder><mrow><mi>i</mi><mi>g</mi><mo stretchy="false">(</mo><msup><mi>γ</mi> <mi>μ</mi></msup><msup><mo stretchy="false">)</mo> <mi>α</mi></msup><msub><mrow></mrow> <mi>β</mi></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mtext>electron-photon</mtext></mrow><mrow><mtext>interaction</mtext></mrow></mfrac></munder><mspace width="thinmathspace"></mspace><mo>.</mo><mspace width="thinmathspace"></mspace><munder><munder><mrow><msub><mover><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>α</mi></mrow></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mtext>incoming</mtext></mrow><mrow><mtext>electron</mtext></mrow></mfrac></mrow><mrow><mtext>propagator</mtext></mrow></mfrac></munder><munder><munder><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mrow><mi>μ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></msub></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mspace width="thinmathspace"></mspace></mrow><mrow><mtext>photon</mtext></mrow></mfrac></mrow><mrow><mtext>propagator</mtext></mrow></mfrac></munder><munder><munder><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mi>β</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></msup></mrow><mo>⏟</mo></munder><mfrac linethickness="0"><mrow><mfrac linethickness="0"><mrow><mtext>outgoing</mtext></mrow><mrow><mtext>electron</mtext></mrow></mfrac></mrow><mrow><mtext>propagator</mtext></mrow></mfrac></munder></mrow><annotation encoding="application/x-tex"> \underset{ \text{loop order} }{ \underbrace{ \hbar^{3/2-1} } } \underset{ \text{electron-photon} \atop \text{interaction} }{ \underbrace{ i g (\gamma^\mu)^\alpha{}_\beta } } \,. \, \underset{ {\text{incoming} \atop \text{electron}} \atop \text{propagator} }{ \underbrace{ \overline{\Delta(-,x)}_{-, \alpha} } } \underset{ { \, \atop \text{photon} } \atop \text{propagator} }{ \underbrace{ G(x,-)_{\mu,-} } } \underset{ { \text{outgoing} \atop \text{electron} } \atop \text{propagator} }{ \underbrace{ \Delta(x,-)^{\beta, -} } } </annotation></semantics></math></div> Hence a typical Feynman diagram in the <a class="existingWikiWord" href="/nlab/show/QED">QED</a> <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> induced by this <a class="existingWikiWord" href="/nlab/show/electron-photon+interaction">electron-photon interaction</a> looks as follows: <center> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramGlobal.jpg" width="560" /> </center> where on the bottom the corresponding <a class="existingWikiWord" href="/nlab/show/Feynman+amplitude">Feynman amplitude</a> <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is shown; now notationally suppressing the contraction of the internal indices and all prefactors. For instance the two solid <a class="existingWikiWord" href="/nlab/show/edges">edges</a> between the <a class="existingWikiWord" href="/nlab/show/vertices">vertices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_12"><semantics><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_13"><semantics><mrow><msub><mi>x</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">x_3</annotation></semantics></math> correspond to the two factors of <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_14"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta(x_2,x_2)</annotation></semantics></math>: <center> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponent1.jpg" width="560" /> </center> This way each sub-graph encodes its corresponding subset of factors in the <a class="existingWikiWord" href="/nlab/show/Feynman+amplitude">Feynman amplitude</a>: <center> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponentTwo.jpg" width="560" /> </center><center> <img src="https://ncatlab.org/nlab/files/FeynmanDiagramComponentThree.jpg" width="560" /> </center> <blockquote> graphics grabbed from <a href="Feynman+diagram#Brouder10">Brouder 10</a> </blockquote> A priori this <a class="existingWikiWord" href="/nlab/show/product+of+distributions">product of distributions</a> is defined away from coincident vertices: <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_15"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>≠</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">x_i \neq x_j</annotation></semantics></math>. The definition at coincident vertices <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0c02e0a3528c2f6e48e15da828bd1c4874be6006_16"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">x_i = x_j</annotation></semantics></math> requires a choice of <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> to the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> locus. This choice is the <a class="existingWikiWord" href="/nlab/show/renormalization">("re-")normalization</a> of the <a class="existingWikiWord" href="/nlab/show/Feynman+amplitude">Feynman amplitude</a>. </div></div> <h2 id="properties">Properties</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/anomalous+magnetic+moment+of+the+electron">anomalous magnetic moment of the electron</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lamb+shift">Lamb shift</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Casimir+effect">Casimir effect</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/axial+anomaly">axial anomaly</a> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fine+structure+constant">fine structure constant</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/photon+propagator">photon propagator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Schwinger+effect">Schwinger effect</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> QED </li> <li> <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a>, <a class="existingWikiWord" href="/nlab/show/quantization+of+Yang-Mills+theory">quantization of Yang-Mills theory</a> </li> </ul> </li> </ul> </li> </ul> <h2 id="references">References</h2> <h3 id="original_articles">Original articles</h3> The term “quantum electrodynamics” is due to <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Dirac">Paul Dirac</a>, The quantum theory of emission and absorption of radiation, Proc. Roy. Soc. London A 114, 243, 1927 (<a href="http://inspirehep.net/record/42586?ln=en">spire</a>, <a href="http://wwwhome.lorentz.leidenuniv.nl/~boyarsky/media/Proc.R.Soc.Lond.-1927-Dirac-243-65.pdf">pdf</a>)</li> </ul> This left some issues with the quantization of the radiation field. These were addressed, and the <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> for the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> was first found in <ul> <li><a class="existingWikiWord" href="/nlab/show/Pascual+Jordan">Pascual Jordan</a>, <a class="existingWikiWord" href="/nlab/show/Wolfgang+Pauli">Wolfgang Pauli</a>, Zur Quantenelektrodynamik ladungsfreier Felder, Zeitschrift für Physik 47, 151 (1928)</li> </ul> A comprehensive, albeit informal, theory of QED and of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> in general was eventually developed by <ul> <li><a class="existingWikiWord" href="/nlab/show/Schwinger-Tomonaga-Feynman-Dyson">Schwinger-Tomonaga-Feynman-Dyson</a>, 1940s</li> </ul> For more on the history see <ul> <li><a class="existingWikiWord" href="/nlab/show/Rudolf+Haag">Rudolf Haag</a>, Early papers on quantum field theory (1992-1930) (<a href="https://link.springer.com/content/pdf/bfm%3A978-3-642-70078-1%2F1%2F1.pdf">pdf</a>)</li> </ul> and (<a href="#Scharf95">Scharf 95, section 0.0</a>). On <a class="existingWikiWord" href="/nlab/show/quantum+electrodynamics">quantum electrodynamics</a> via <a class="existingWikiWord" href="/nlab/show/discretized+light-cone+quantization">discretized light-cone quantization</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Andrew+C.+Tang">Andrew C. Tang</a>, Discretized light cone quantization: Application to quantum electrodynamics, PhD thesis, Stanford (1990) [<a href="https://inspirehep.net/literature/296776">spire:296776</a>, <a href="https://inspirehep.net/files/5ef318e56c87e7561d36559f54895bd1">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Tang-DLCQ.pdf" title="pdf">pdf</a>] </li> <li id="TangBrodskyPauli91"> <a class="existingWikiWord" href="/nlab/show/Andrew+C.+Tang">Andrew C. Tang</a>, <a class="existingWikiWord" href="/nlab/show/Stanley+J.+Brodsky">Stanley J. Brodsky</a>, <a class="existingWikiWord" href="/nlab/show/Hans-Christian+Pauli">Hans-Christian Pauli</a>, Discretized light-cone quantization: Formalism for quantum electrodynamics, Phys. Rev. D 44 (1991) 1842 [<a href="https://doi.org/10.1103/PhysRevD.44.1842">doi:10.1103/PhysRevD.44.1842</a>] </li> </ul> Discussion of <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> in <a class="existingWikiWord" href="/nlab/show/QED">QED</a>: <ul> <li id="Scharf96"> <a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Scharf">Günter Scharf</a>, Vacuum stability in quantum field theory, Nuovo Cim. A109 (1996) 1605-1607 (<a href="http://inspirehep.net/record/432208/">spire:432208</a>) </li> <li id="Nikolic01"> <a class="existingWikiWord" href="/nlab/show/Hrvoje+Nikoli%C4%87">Hrvoje Nikolić</a>, Physical stability of the QED vacuum, 2001 (<a href="https://arxiv.org/abs/hep-ph/0105176">arXiv:hep-ph/0105176</a>) </li> </ul> The rigorous construction of <a class="existingWikiWord" href="/nlab/show/perturbative+QFT">perturbative</a> QED in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a> is worked out in <ul> <li><a href="#Scharf95">Scharf 95</a></li> </ul> and refined to <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> in <ul> <li id="DuetschFredenhagen98"><a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, A local (perturbative) construction of observables in gauge theores: the example of QED , Commun. Math. Phys. 203 (1999), no.1, 71-105, (<a href="http://xxx.uni-augsburg.de/abs/hep-th/9807078">arXiv:hep-th/9807078</a>).</li> </ul> and generalized to possibly non-abelian <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> in <ul> <li id="Hollands07"><a class="existingWikiWord" href="/nlab/show/Stefan+Hollands">Stefan Hollands</a>, Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys. 20:1033-1172, 2008 (<a href="https://arxiv.org/abs/0705.3340">arXiv:0705.3340</a>)</li> </ul> The <a class="existingWikiWord" href="/nlab/show/weak+adiabatic+limit">weak adiabatic limit</a> of QED was established in <ul> <li id="BlanchardSeneor75"> P. Blanchard and R. Seneor, Green’s functions for theories with massless particles (in perturbation theory), Ann. Inst. H. Poincaré Sec. A 23 (2), 147–209 (1975) (<a href="http://www.numdam.org/item?id=AIHPA_1975__23_2_147_0">Numdam</a>) </li> <li id="Duch17"> <a class="existingWikiWord" href="/nlab/show/Pawe%C5%82+Duch">Paweł Duch</a>, Massless fields and adiabatic limit in quantum field theory (<a href="https://arxiv.org/abs/1709.09907">arXiv:1709.09907</a>) </li> </ul> The <a class="existingWikiWord" href="/nlab/show/local+net+of+algebras+of+observables">local net of algebras of observables</a> and hence the <a class="existingWikiWord" href="/nlab/show/algebraic+adiabatic+limit">algebraic adiabatic limit</a> was worked out in <ul> <li><a href="#DuetschFredenhagen98">Dütsch-Fredenhagen 98</a></li> </ul> Further discussion of the <a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a> and <a class="existingWikiWord" href="/nlab/show/infrared+divergences">infrared divergences</a> includes <ul> <li>Daniel Kapec, Malcolm Perry, Ana-Maria Raclariu, <a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>, Infrared Divergences in QED, Revisited, Phys. Rev. D 96, 085002 (2017) (<a href="https://arxiv.org/abs/1705.04311">arXiv:1705.04311</a>)</li> </ul> <h3 id="review">Review</h3> Traditional discussion includes <ul> <li> <a class="existingWikiWord" href="/nlab/show/Radovan+Dermisek">Radovan Dermisek</a>, Quantum Electrodynamics (QED) (<a href="http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-5-4p.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DermisekQED.pdf" title="pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Hitoshi+Murayama">Hitoshi Murayama</a>, Quantum Electrodynamics (<a href="http://hitoshi.berkeley.edu/221B-S02/QED.pdf">pdf</a>) </li> <li id="Zeidler"> <a class="existingWikiWord" href="/nlab/show/Eberhard+Zeidler">Eberhard Zeidler</a>, Quantum field theory II: Quantum electrodynamics – A bridge between mathematicians and physicists, Springer (2009) </li> <li> Einan Gardi, lectures 19, 20 of <a href="https://www2.ph.ed.ac.uk/~egardi/MQFT/">Modern Quantum Field Theory</a>, 2015 (<a href="https://www2.ph.ed.ac.uk/~egardi/MQFT/MQFT_2015_lecture_19_20.pdf">pdf</a>) </li> </ul> Mathematically rigorous discussion in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> is in <ul> <li id="Scharf95"> <a class="existingWikiWord" href="/nlab/show/G%C3%BCnter+Scharf">Günter Scharf</a>, <a class="existingWikiWord" href="/nlab/show/Finite+Quantum+Electrodynamics+--+The+Causal+Approach">Finite Quantum Electrodynamics – The Causal Approach</a>, Berlin: Springer-Verlag, 1995, 2nd edition </li> <li id="Duetsch18"> <a class="existingWikiWord" href="/nlab/show/Michael+D%C3%BCtsch">Michael Dütsch</a>, chapter 5 of <a class="existingWikiWord" href="/nlab/show/From+classical+field+theory+to+perturbative+quantum+field+theory">From classical field theory to perturbative quantum field theory</a>, 2018 following (<a href="#DuetschFredenhagen98">Dütsch-Fredenhagen 98</a>) </li> </ul> <h3 id="phenomenology">Phenomenology</h3> Comparison to <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a> is reviewed in <ul> <li>Tests of QED (<a href="http://edu.itp.phys.ethz.ch/hs10/ppp1/PPP1_6.pdf">pdf</a>)</li> </ul> Specifically via the <a class="existingWikiWord" href="/nlab/show/Lamb+shift">Lamb shift</a>: <ul> <li>Savely G. Karshenboim, D. I. Mendeleev, The Lamb Shift of Hydrogen and Low-Energy Tests of QED (<a href="https://arxiv.org/abs/hep-ph/9411356">arXiv:hep-ph/9411356</a>)</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on November 24, 2023 at 12:20:36. 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